Extensions for Rare Events

In the following sections we discuss some extensions which were mentioned in the previous chapters. Of main interest is the extension to general rare events in relation to a random sequence applying the same method as used for dealing with exceedances. In addition we treat now also the point process of all exceedances if clustering occurs. These results are applied to the processes of peaks over a threshold and of rare events. Finally, in the same way general rare events are considered without relation to a random sequence. Here triangular arrays of rare events will be analyzed by the same approach. This extension unifies easily the different local dependence conditions. Its application to multivariate extremes is then straightforward. As a particular case triangular arrays of rare events in relation with exceedances of Gaussian sequences are considered since they are basic for maxima of a continuous Gaussian process. This analysis reveals also a new definition of Pickands constants.